91Ó°ÊÓ

True or False. a. If the \(p\) -value for a test is .036 , the null hypothesis can be rejected at the \(\alpha=.05\) level of significance. b. In a formal test of hypothesis, \(\alpha\) is the probability that the null hypothesis is incorrect. c. If the \(p\) -value is very small for a test to compare two population means, the difference between the means must be large. d. Power \(\left(\theta^{*}\right)\) is the probability that the null hypothesis is rejected when \(\theta=\theta^{*}\). e. Power( \((\theta)\) is always computed by assuming that the null hypothesis is true. f. If \(.01 < p\) -value \( < .025\), the null hypothesis can always be rejected at the \(\alpha=.02\) level of significance. g. Suppose that a test is a uniformly most powerful \(\alpha\) -level test regarding the value of a parameter \(\theta .\) If \(\theta_{a}\) is a value in the alternative hypothesis, \(\beta\left(\theta_{a}\right)\) might be smaller for some other \(\alpha\) -level test. h. When developing a likelihood ratio test, it is possible that \(L\left(\widehat{\Omega}_{0}\right) > L(\widehat{\Omega})\) i. \(-2 \ln (\lambda)\) is always positive.

Step by step solution

01

Analyze Statement a

For statement a: The null hypothesis can be rejected at the \( \alpha = 0.05 \) level of significance if the \( p \)-value is less than \( 0.05 \). Since the \( p \)-value is 0.036, which is less than 0.05, the null hypothesis can be rejected. Thus, the statement is true.

02

Analyze Statement b

For statement b: \( \alpha \) is actually the probability of rejecting the null hypothesis when it is true (Type I error), not the probability that the null hypothesis is incorrect. Thus, this statement is false.

03

Analyze Statement c

For statement c: A very small \( p \)-value indicates that the observed data is unlikely under the null hypothesis, but it does not necessarily mean the difference between the means must be large. Thus, this statement is false.

04

Analyze Statement d

For statement d: Power is the probability of rejecting the null hypothesis given that a specific alternative hypothesis \( \theta^{*} \) is true. Hence, the statement is correctly describing power, so it is true.

05

Analyze Statement e

For statement e: Power is typically computed under the assumption that the alternative hypothesis is true, not the null hypothesis. Therefore, the statement is false.

06

Analyze Statement f

For statement f: If \( 0.01 < p \text{-value} < 0.025 \), then the \( p \)-value is less than \( \alpha = 0.02 \), so the null hypothesis can be rejected at this level. Thus, the statement is true.

07

Analyze Statement g

For statement g: A uniformly most powerful test means it has the smallest \( \beta \) (Type II error) for given \( \alpha \), but it could still be possible for another test to have a smaller \( \beta \) for specific points under different setups but not uniformly. Thus, this statement is false.

08

Analyze Statement h

For statement h: In a likelihood ratio test, \( L(\widehat{\Omega}_{0}) \leq L(\widehat{\Omega}) \) by definition, so the statement is false.

09

Analyze Statement i

For statement i: \(-2 \ln(\lambda)\) being positive depends on the value of the likelihood ratio for a specific test; it is not always positive. Therefore, the statement is false.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

p-value interpretation

The p-value is a crucial part of hypothesis testing. It helps us decide whether the evidence in our data is strong enough to reject the null hypothesis. A smaller p-value indicates stronger evidence against the null hypothesis. If the p-value is less than the significance level (often denoted by \( \alpha \)), we reject the null hypothesis. For example, if \( \alpha = 0.05 \) and the p-value is 0.036, as in the original exercise, it suggests that the results are statistically significant, and the null hypothesis can be rejected. This is because 0.036 is less than 0.05. In summary, the p-value helps quantify the evidence and make decisions based on statistical data.

Type I error

In hypothesis testing, a Type I error occurs when we reject a true null hypothesis. It is also known as a "false positive" error. The probability of making a Type I error is represented by the significance level \( \alpha \). This means that if we set \( \alpha = 0.05 \), there is a 5% risk of rejecting the null hypothesis when it is actually true. Controlling the Type I error rate is important for ensuring the validity of a hypothesis test. Reducing the value of \( \alpha \) can decrease the likelihood of making a Type I error, but it might increase the chance of a Type II error, which involves failing to reject a false null hypothesis. Therefore, balancing these errors is crucial in statistical analysis.

Statistical power

Statistical power is the probability that a test correctly rejects a false null hypothesis. High statistical power means a higher chance of detecting an effect when there is one. The power of a test is influenced by several factors, including the sample size, effect size, significance level, and variability of the data. For instance, increasing the sample size generally increases power because it reduces variability. Moreover, choosing a larger significance level \( \alpha \) can also enhance power, but this might lead to a higher risk of a Type I error. In the context of the original exercise, the power is specifically defined as the probability that the null hypothesis is rejected when a particular alternative hypothesis \( \theta^{*} \) is true. Knowing the power of a test helps researchers understand the likelihood that their test will detect an actual effect.

Likelihood ratio test

The likelihood ratio test is a statistical test used to compare the fit of two models: a null model and an alternative model. It measures how well the data support one model over the other by comparing their likelihoods. The key statistic in this test is the likelihood ratio, usually expressed as \(-2 \ln(\lambda)\), where \( \lambda \) is the ratio of the likelihoods of the two models. In practice, the likelihood ratio test helps determine whether a more complex model provides a significantly better fit to the data than the simpler model. However, as indicated in the original exercise, the statistic \(-2 \ln(\lambda)\) is not always positive and depends on the particular situation being analyzed. Understanding likelihood ratio tests is essential for researchers who wish to compare different models and make decisions about the best approach based on their data.